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聚类中心","从所有样本中随机选取k样本作为初始的聚类中心","从机器学习的角度来看，信息熵表示的是信息量的期望值。如果数据集中的数据需要被分成多个类别，则信息量","从这个公式也可以看出，如果概率是0或者是1的时候，熵就是0。（因为这种情况下随机变量的不确定性是最低的），那如果概率是0.5也就是五五开的时候，此时熵达到最大，也就是1。（就像扔硬币，你永远都猜不透你下次扔到的是正面还是反面，所以它的不确定性非常高）。所以呢，熵越大，不确定性就越高。","但是，并不是每个青少年都符合这个公式，只能说每个青少年的身高体重都存在这么一种近似的线性关系。这就是其实就是简单的线性回归，那么，到底什么是线性回归呢？假如我们将青少年的身高和体重值作为坐标，不同人的身高体重就会在平面上构成不同的坐标点，然后用一条直线，尽可能的去拟合这些点，这就是简单的线性回归。","低","体重/kg","何为最近","例如：要做房价预测，每平方是万元，我们预测结果也是万元。那么差值的平方单位应该是千万级别的。那我们不太好描述自己做的模型效果。怎么说呢？我们的模型误差是多少千万？于是干脆就开个根号就好了。我们误差的结果就跟我们数据是一个级别的了，在描述模型的时候就说，我们模型的误差是多少万元。","信息增益","信息是个很抽象的概念。人们常常说信息很多，或者信息较少，但却很难说清楚信息到底有多少。比如一本五十万字的中文书到底有多少信息量。","信息熵","值越大代表用户越喜欢某种元素。","值越大代表电影中某元素内容越多。","假如要算性别和活跃度这两个特征的信息增益的话，首先要先算总的熵和条件熵。总的熵其实非常好算，就是把标签作为随机变量x。上表中标签只有两种（0和1）因此随机变量x的取值只有0或者1。所以要计算熵就需要先分别计算标签为0的概率和标签为1的概率。从表中能看出标签为0的数据有10条，所以标签为0的概率等于2/3。标签为1的概率为1/3。所以熵为：","假设我们有k个簇：(c1,c2,...,ck)(c_1,c_2,...,c_k)(c1,c2,...,ck)","假设现在水果店里有3个西瓜，它们的属性如下：","假设电影评分表y（为m行n列的矩阵）,我们考虑d种元素，则电影评分表可以分解为用户喜好表x（为m行d列的矩阵），与电影内容表w（为d行n列的矩阵）。其中d为超参数，大小由我们自己定。","关于何为最近，大家应该自然而然就会想到可以用两个样本之间的距离大小来衡量，我们常用的有两种距离：","其中p表示预测值，y表示真实值，m为样本总个数，i表示第i个样本。最后，我们再使用正规方程解来求得我们所需要的参数。","其中x0=1,w0=b,通过向量化公式可写成如下形式：","其中xix_ixi表示多个类别中的第i个类别，p(xi)p(x_i)p(xi)表示概率：","其中x表示特征值(如：体重值)，w表示权重，b表示偏置，y表示标签(如：身高值)。","其中yi表示第i个样本的真实标签，pi表示第i个样本的预测标签。线性回归的目的就是让损失函数最小。那么，模型训练出来了，我们再测试集上用损失函数来评估也是可以的。","其中每一行代表一个鸢尾花样本各个属性的值。","其中，record为评分记录矩阵。","其中，uiu_iui为质心，表达式为：","其中，xix_ixi表示第i个特征，wiw_iwi表示第i个特征对于的权重，b表示偏置，y表示标签。","其中，基准模型值的随机瞎猜的模型。","其中，标签y为m行1列的矩阵，训练特征x为m行(n+1)列的矩阵，回归系数w为(n+1)行1列的矩阵，对w求导，并令其导数为零可解得：","其中：","其实分子表示的是模型预测时产生的误差，分母表示的是对任意样本都预测为所有标签均值时产生的误差，由此可知：","其损失函数可以表示为","再根据测试集标签即真实分类结果，计算出正确率：","再根据测试集标签，可以计算出正确率：","决策树","决策树是一种可以用于分类与回归的机器学习算法，但主要用于分类。用于分类的决策树是一种描述对实例进行分类的树形结构。决策树由结点和边组成，其中结点分为内部结点和叶子结点，内部结点表示一个特征或者属性，叶子结点表示标签。","决策树说通俗点就是一棵能够替我们做决策的树，或者说是我们人类在要做决策时脑回路的一种表现形式，我们可以从下面这个例子来了解决策树是什么。","则我们的目的就是使的簇内的每个点到簇的质心的距离最小，即最小化平方误差mse：","则梯度为：","前言","加权投票","可以发现，使用实现的方法进行聚类的结果与真实情况非常吻合。","可以看到，使用knn对手写数字进行识别，正确率能达到99%以上。","可以看到，使用决策树对鸢尾花进行分类，正确率可以达到100%","可视化数据分布：","可视化结果：","同样的只需要调用之前实现线性回归方法就可以对测试集的波士顿房价数据进行预测了：","否","因素1","因素2","图b:假设k=2，我们最开始先随机初始2个质心(红色与蓝色的点)。","图c:计算每个样本到两个质心的距离，并将其归为与其距离最近的质心那个簇。","图d:更新质心，我们可以看到，红色与蓝色的点位置有了变化。","图e:重新计算样本到质心距离，并重新划分样本属于哪个簇。","图f:直到质心位置变换小于阈值，停止迭代。","在实际的场景中，我们可能需要研究数据集中某个特征等于某个值时的信息熵等于多少，这个时候就需要用到条件熵。条件熵h(y|x)表示特征x为某个值的条件下，类别为y的熵。条件熵的计算公式如下：","在推荐系统中，我们经常看到如下图的表格，表格中的数字代表用户对某个物品的评分，0代表未评分。我们希望能够预测目标用户对物品的评分，进而根据评分高低，将分高的物品推荐给用户。","在炎热的夏天，没有什么比冰镇后的西瓜更能令人感到心旷神怡的了。现在我要去水果店买西瓜，但怎样我才会买这个西瓜呢？那么，有可能我会有以下这个决策过程：","在生活中，我们常常能碰到这么一种情况，一个变量会跟着另一个变量的变化而变化，如圆的周长与半径的关系，当圆的半径确定了，那么周长也就确定了。还有一种情况就是，两个变量之间看似存在某种关系，但又没那么确定，如青少年的身高与体重，他们存在一种近似的线性关系：","均值","均值算法原理","均值算法思想","均值算法流程","均值）聚类，之所以称为","基于上面的设想，我们只要知道所有用户对电影内容各种元素喜欢程度与所有电影内容的成分，就能预测出所有用户对所有电影的评分了。","基于矩阵分解的协同过滤算法思想为：一个用户评分矩阵可以分解为一个用户喜好矩阵与内容矩阵，我们只要能找出正确的用户喜好矩阵参数与内容矩阵参数（即表内的值），就能对用户评分进行预测，再根据预测结果对用户进行推荐。","基于矩阵分解的协同过滤算法正好能解决这个问题。","基于矩阵分解的协同过滤算法通常都会构造如下图所示评分表y，这里我们以电影为例：","多元线性回归","够不够冰","大家已经知道，要使用基于矩阵分解的协同过滤算法，首先得有用户与电影评分的矩阵，而我们实际中的数据并不是以这样的形式保存，所以在使用算法前要先构造出用户","女","如:","如上图，当k=3时离绿色的圆最近的三个样本中，有两个红色的三角形，一个蓝色的正方形，则此时绿色的圆应该分为红色的三角形这一类。而当k=5时，离绿色的圆最近的五个样本中，有两个红色的三角形，三个蓝色的正方形，则此时绿色的圆应该分为蓝色的正方形这一类。","如上图，虽然蓝色正方形与红色三角形数量一样，但是根据加权投票的规则，绿色的圆应该属于蓝色正方形这个类别。","如果有两个类型的样本数一样且最多，那么最终该样本应该属于哪个类型","如果聚类中心几乎没有变化，说明算法已经收敛，退出迭代","对中心进行更新","对于分类问题，我们可以使用正确率来衡量模型的性能好坏，很明显，回归问题并不能使用正确率来衡量，那么，回归问题可以使用哪些指标用来评估呢？","对于所有用户，我们可以将矩阵x与矩阵w相乘，得到所有用户对所有电影的预测评分如下表：","对整个数据集x进行kmeans聚类，返回其聚类的标签","对每个参数求得偏导如下：","对线性回归模型，假设训练集中m个训练样本，每个训练样本中有n个特征，可以使用矩阵的表示方法，预测函数可以写为：","将所有样本进行归类，其所在的类别的索引就是其类别标签","将所有样本进行归类，归类规则就是将该样本归类到与其最近的中心","将所有进行归类，归类规则就是将该样本归类到与其最近的中心","将用户喜好矩阵与内容矩阵进行矩阵乘法就能得到用户对物品的预测结果，而我们的目的是预测结果与真实情况越接近越好。所以，我们将预测值与评分表中已评分部分的值构造平方差损失函数：","就好比，我在玩读心术。你心里想一件东西，我来猜。我已开始什么都没问你，我要猜的话，肯定是瞎猜。这个时候我的熵就非常高。然后我接下来我会去试着问你是非题，当我问了是非题之后，我就能减小猜测你心中想到的东西的范围，这样其实就是减小了我的熵。那么我熵的减小程度就是我的信息增益。","当然，每一个样本就是一个数字，我们可以把它还原为8x8的大小进行查看：","得到指定特征列的值的集合","性别","性别为女的熵为：","性别为男的熵为：","性别的信息增益=总的熵","想要直接求得最小值是非常困难的，通常我们使用启发式的迭代方法，过程如下图：","我们可以使用sklearn直接对数据进行加载，代码如下：","我们可以先根据数据的真实标签查看数据类别情况：","我们可以根据求得的预测值，计算出mse值与r","我们可以直接使用sklearn直接对数据进行加载，代码如下：","我们已经知道线性回归模型如下：","我们已经知道，如何判别一个样本属于哪个类型，主要是看离它最近的几个样本中哪个类型的数量最多，则该样本属于数量最多的类型。这里，存在两个问题：","我们已经知道，构造一棵决策树其实就是根据数据的特征(内部节点)对数据一步一步的进行划分，从而达到分类的目的。但是，每一步根据哪个特征来进行划分呢？这个就是构造决策树的关键。其实构造决策树时会遵循一个指标，有的是按照信息增益来构建，如id3算法；有的是信息增益率来构建，如c4.5算法；有的是按照基尼系数来构建的，如cart算法。但不管是使用哪种构建算法，决策树的构建过程通常都是一个递归选择最优特征，并根据特征对训练集进行分割，使得对各个子数据集有一个最好的分类的过程。这里我们以id3算法为例，详细介绍构建决策树相关知识。","我们最终的目的是根据创建的决策树模型对测试集数据进行预测，算法实现流程如下：","我们最终的目的是通过训练出来的线性回归模型对测试集数据进行预测，算法实现流程如下：","我们的目的就是最小化平方差损失函数，通常机器学习都是使用梯度下降的方法来最小化损失函数得到正确的参数。","我们认为，有很多因素会影响到用户给电影评分，如电影内容：感情戏，恐怖元素，动作成分，推理悬疑等等。假设我们现在想预测用户2对电影2的评分，用户2他很喜欢看动作片与推理悬疑，不喜欢看感情戏与恐怖的元素，而电影2只有少量的感情戏与恐怖元素，大部分都是动作与推理的剧情，则用户2对电影2评分可能很高，比如5分。","所以信息增益如果套上机器学习的话就是，如果把特征a对训练集d的信息增益记为g(d,","手写数字数据","手写数字数据集一共有1797个样本，每个样本有64个特征。每个特征的值为0","拿到bestfeature的所有特征值","损失函数python实现代码如下：","接下来就只需要调用之前实现的knn_clf方法就可以对测试集的手写数字进行识别了：","接下来就是条件熵的计算，以性别为男的熵为例。表格中性别为男的数据有8条，这8条数据中有3条数据的标签为1，有5条数据的标签为0。所以根据条件熵的计算公式能够得出该条件熵为：","推荐系统","提取码：ve3v","效果如下：","数据下载连接","数据挖掘其实就是从数据中学习到规律，再将学习到的规律对未知的数据进行分析。数据的质量直接影响到模型学习的好坏，而我们最开始获取的数据其中绝大多数是“有毛病”的，不利于后期进行分析。所以我们在分析前需要进行数据的预处理。","数据探索","数据集中部分数据与标签如下图所示：","数据集中部分数据如下所示：","数据集中部分标签如下图所示：","数据预处理","是","是否便宜","是否有籽","是否流失","是因为它可以发现k个簇，且每个簇的中心采用簇中所含值的均值计算而成。簇内的样本连接紧密，而簇之间的距离尽量大。简单来讲，其思想就是物以类聚。","曼哈顿距离：顾名思义，在曼哈顿街区要从一个十字路口开车到另一个十字路口，驾驶距离显然不是两点间的直线距离。这个实际驾驶距离就是“曼哈顿距离”。曼哈顿距离也称为“城市街区距离”。","最后，使用我们实现的k","本次使用电影评分数据为672个用户对9123部电影的评分记录，部分数据如下：","本次我们使用的仍然是鸢尾花数据，不过为了能够进行可视化我们只使用数据中的两个特征：","条件熵","构建对应特征值的子样本集sub_feature,","构造出表格后，我们就能使用上一关实现的方法来对用户进行电影推荐了：","构造用户","标签","标签中的值0,1,2分别代表鸢尾花三种不同的类别。","样本中只有一个特征或者所有样本的特征都一样的话就看哪个label的票数高","样本里都是同一个label没必要继续分叉了","根据上述的计算方法可知，总熵为：","根据信息增益拿到特征的索引","梯度python代码如下：","欧氏距离：欧氏距离是最容易直观理解的距离度量方法，我们小学、初中和高中接触到的两个点在空间中的距离一般都是指欧氏距离。","正规方程解","波士顿房价数据","波士顿房价数据集共有506条房价数据，每条数据包括对指定房屋的13项数值型特征和目标房价组成。我们需要通过数据特征来对目标房价进行预测。","活跃度","活跃度为中的熵为：","活跃度为低的熵为：","活跃度为高的熵为：","活跃度的信息增益=总的熵","然后再对数据集进行划分：","然后再进行梯度下降:","然后我们再使用实现的决策树分类方法就可以对测试集数据进行分类：","然后我们划分出训练集与测试集，训练集用来训练模型，测试集用来检测模型性能。代码如下：","然后，我们还有电影编号与电影名字对应的数据如下：","物品1","物品2","物品3","物品4","物品5","现在已经知道了什么是熵，什么是条件熵。接下来就可以看看什么是信息增益了。所谓的信息增益就是表示我已知条件x后能得到信息y的不确定性的减少程度。","现在有了总的熵和条件熵之后就能算出性别和活跃度这两个特征的信息增益了。","瓤是否够红","用一句话来总结knn算法的思想就是近朱者赤近墨者黑。","用户1","用户2","用户2对电影2评分为：5×1.0+0×0.01=5.05\\time","用户3","用户4","用户喜好表x：","由于信息熵是信息量的期望值，所以信息熵h(x)h(x)h(x)的定义如下(其中n为数据集中类别的数量)：","电影1","电影2","电影3","电影4","电影5","电影内容表：w:","电影评分数据","电影评分矩阵","电影评分矩阵，python实现代码如下：","男","直到1948年，香农提出了“信息熵”的概念，才解决了对信息的量化度量问题。信息熵这个词是香农从热力学中借用过来的。热力学中的热熵是表示分子状态混乱程度的物理量。香农用信息熵的概念来描述信源的不确定度。信源的不确定性越大，信息熵也越大。","看看分类算法的衡量标准就是正确率，而正确率又在0～1之间，最高百分之百。最低0。那么线性回归有没有这样的衡量标准呢？r","第一章","第一行数据表示用户1对电影30评分为2.5分。","第七章","第三章","第九章","第二章","第二行数据表示用户7对电影30评分为3分。","第五章","第八章","第六章","第十章","第四章","简单的线性回归模型如下：","简单线性回归","简单线性回归中，一个变量跟另一个变量的变化而变化，但是生活中，还有很多变量，可能由多个变量的变化决定着它的变化，比如房价，影响它的因素可能有：房屋面积、地理位置等等。如果我们要给它们建立出近似的线性关系，这就是多元线性回归，多元线性回归模型如下：","线性回归","线性回归python实现代码如下：","线性回归模型训练流程图如下：","线性回归训练流程","绪论","编号","而我们的目的就是找出能够正确预测的多元线性回归模型，即找出正确的w（即权重与偏置）。那么如何寻找呢？通常在监督学习里面都会使用这么一个套路，构造一个损失函数，用来衡量真实值与预测值之间的差异，然后将问题转化为最优化损失函数。既然损失函数是用来衡量真实值与预测值之间的差异那么很多人自然而然的想到了用所有真实值与预测值的差的绝对值来表示损失函数。不过带绝对值的函数不容易求导，所以采用mse(均方误差)作为损失函数，公式如下：","花瓣宽度","花瓣长度","花萼宽度","花萼长度","若只考虑两种元素则用户喜好表与电影内容表如下：","获得信息增益最高的特征","衡量线性回归的性能指标","计算一个样本与数据集中所有样本的欧氏距离的平方","计算信息增益","计算新的聚类中心","计算条件熵","计算标签在数据集中出现的概率","计算熵","距离度量","身高/cm","近邻","近邻算法原理","近邻算法思想","近邻算法流程","近邻（k","返回距离该样本最近的一个中心索引[0,","这个就是正规方程解，我们可以通过正规方程解直接求得我们所需要的参数。","这里使用python实现了mse，r","进行分类","进行聚类","进行识别","进行预测","迭代，直到算法收敛(上一次的聚类中心和这一次的聚类中心几乎重合)或者达到最大迭代次数","递归构建决策树","那么根据我的决策过程我会买1和2号西瓜。这个帮助我选择西瓜的过程，就是一个决策树。由之前介绍的知识可以知道，黄色部分为内部节点，蓝色部分为叶子节点。","那么误差单位就是万元。数子可能是3，4，5之类的。那么预测身高就可能是0.1，0.6之类的。没有什么可读性，到底多少才算好呢？不知道，那要根据模型的应用场景来。","那信息增益算出来之后有什么意义呢？回到读心术的问题，为了我能更加准确的猜出你心中所想，我肯定是问的问题越好就能猜得越准！换句话来说我肯定是要想出一个信息增益最大（减少不确定性程度最高）的问题来问你。其实id3算法也是这么想的。id3算法的思想是从训练集d中计算每个特征的信息增益，然后看哪个最大就选哪个作为当前结点。然后继续重复刚刚的步骤来构建决策树。","高","鸢尾花数据","鸢尾花数据集是一类多重变量分析的数据集，一共有150个样本，通过花萼长度，花萼宽度，花瓣长度，花瓣宽度","（mean"],"pipeline":["stopWordFilter","stemmer"]},"store":{"./":{"url":"./","title":"前言","keywords":"","body":"\n"},"Chapter1/":{"url":"Chapter1/","title":"第一章 绪论","keywords":"","body":"第一章 绪论\n"},"Chapter1/为什么要数据挖掘.html":{"url":"Chapter1/为什么要数据挖掘.html","title":"1.1:为什么要数据挖掘","keywords":"","body":"1.1:为什么要数据挖掘\n"},"Chapter1/什么是数据挖掘.html":{"url":"Chapter1/什么是数据挖掘.html","title":"1.2: 什么是数据挖掘","keywords":"","body":"1.2: 什么是数据挖掘\n"},"Chapter1/数据挖掘主要任务.html":{"url":"Chapter1/数据挖掘主要任务.html","title":"1.3:数据挖掘主要任务","keywords":"","body":"1.3:数据挖掘主要任务\n"},"Chapter2/":{"url":"Chapter2/","title":"第二章 数据探索","keywords":"","body":"第二章 数据探索\n"},"Chapter2/数据与属性.html":{"url":"Chapter2/数据与属性.html","title":"2.1:数据与属性","keywords":"","body":"2.1:数据与属性\n"},"Chapter2/数据的基本统计指标.html":{"url":"Chapter2/数据的基本统计指标.html","title":"2.2:数据的基本统计指标","keywords":"","body":"2.2:数据的基本统计指标\n"},"Chapter2/数据可视化.html":{"url":"Chapter2/数据可视化.html","title":"2.3:数据可视化","keywords":"","body":"2.3:数据可视化\n"},"Chapter2/相似性度量.html":{"url":"Chapter2/相似性度量.html","title":"2.4:相似性度量","keywords":"","body":"2.4:相似性度量\n"},"Chapter3/":{"url":"Chapter3/","title":"第三章 数据预处理","keywords":"","body":"第三章 数据预处理\n"},"Chapter3/为什么要数据预处理.html":{"url":"Chapter3/为什么要数据预处理.html","title":"3.1:为什么要数据预处理","keywords":"","body":"3.1:为什么要数据预处理\n数据挖掘其实就是从数据中学习到规律，再将学习到的规律对未知的数据进行分析。数据的质量直接影响到模型学习的好坏，而我们最开始获取的数据其中绝大多数是“有毛病”的，不利于后期进行分析。所以我们在分析前需要进行数据的预处理。\nsklearn.preprocessing包提供了几个常用的函数和转换类型，用它们将一个原始的特征向量转化为一个更适于数据分析的表示形式。一般来说，学习算法收益于数据集的标准形式。如果数据中存在异常点，稳健的数据规范或转换是更适合的。\n"},"Chapter3/标准化.html":{"url":"Chapter3/标准化.html","title":"3.2:标准化","keywords":"","body":"3.2:标准化\n"},"Chapter3/非线性变换.html":{"url":"Chapter3/非线性变换.html","title":"3.3:非线性变换","keywords":"","body":"3.3:非线性变换\n"},"Chapter3/归一化.html":{"url":"Chapter3/归一化.html","title":"3.4:归一化","keywords":"","body":"3.4:归一化\n"},"Chapter3/离散值编码.html":{"url":"Chapter3/离散值编码.html","title":"3.5:离散值编码","keywords":"","body":"3.5:离散值编码\n"},"Chapter3/生成多项式特征.html":{"url":"Chapter3/生成多项式特征.html","title":"3.6:生成多项式特征","keywords":"","body":"3.6:生成多项式特征\n"},"Chapter3/估算缺失值.html":{"url":"Chapter3/估算缺失值.html","title":"3.7:估算缺失值","keywords":"","body":"3.7:估算缺失值\n"},"Chapter4/":{"url":"Chapter4/","title":"第四章 k-近邻","keywords":"","body":"第四章 k-近邻\n"},"Chapter4/k-近邻算法思想.html":{"url":"Chapter4/k-近邻算法思想.html","title":"4.1:k-近邻算法思想","keywords":"","body":"4.1:k-近邻算法思想\nk-近邻（k-nearest neighbor ,knn）是一种分类与回归的方法。我们这里只讨论用来分类的knn。所谓k最近邻，就是k个最近的邻居的意思，说的是每个样本都可以用它最近的k个邻居来代表。\nknn算法的核心思想是如果一个样本在特征空间中的k个最相邻的样本中的大多数属于某一个类别，则该样本也属于这个类别，并具有这个类别上样本的特性。该方法在确定分类决策上只依据最邻近的一个或者几个样本的类别来决定待分样本所属的类别。knn方法在类别决策时，只与极少量的相邻样本有关。\n\n如上图，当k=3时离绿色的圆最近的三个样本中，有两个红色的三角形，一个蓝色的正方形，则此时绿色的圆应该分为红色的三角形这一类。而当k=5时，离绿色的圆最近的五个样本中，有两个红色的三角形，三个蓝色的正方形，则此时绿色的圆应该分为蓝色的正方形这一类。\n用一句话来总结knn算法的思想就是近朱者赤近墨者黑。\n"},"Chapter4/k-近邻算法原理.html":{"url":"Chapter4/k-近邻算法原理.html","title":"4.2:k-近邻算法原理","keywords":"","body":"4.2:k-近邻算法原理\n我们已经知道，如何判别一个样本属于哪个类型，主要是看离它最近的几个样本中哪个类型的数量最多，则该样本属于数量最多的类型。这里，存在两个问题：\n\n何为最近\n如果有两个类型的样本数一样且最多，那么最终该样本应该属于哪个类型\n\n距离度量\n关于何为最近，大家应该自然而然就会想到可以用两个样本之间的距离大小来衡量，我们常用的有两种距离：\n\n欧氏距离：欧氏距离是最容易直观理解的距离度量方法，我们小学、初中和高中接触到的两个点在空间中的距离一般都是指欧氏距离。\n\n\n二维平面上欧式距离计算公式：\nd12=(x1(1)−x1(2))2+(x2(1)−x2(2))2\rd_{12} = \\sqrt{(x^{(1)}_1-x^{(2)}_1)^2+(x^{(1)}_2-x^{(2)}_2)^2}\rd12=√(x1(1)−x1(2))2+(x2(1)−x2(2))2\nn维平面上欧氏距离计算公式：\nd12=∑i=1n(xi(1)−xi(2))2\rd_{12}=\\sqrt{\\sum\\limits_{i=1}^n(x^{(1)}_i-x^{(2)}_i)^2}\rd12=√i=1∑n(xi(1)−xi(2))2\n\n曼哈顿距离：顾名思义，在曼哈顿街区要从一个十字路口开车到另一个十字路口，驾驶距离显然不是两点间的直线距离。这个实际驾驶距离就是“曼哈顿距离”。曼哈顿距离也称为“城市街区距离”。\n\n\n二维平面上曼哈顿距离计算公式：\nd12=∣x1(1)−x1(2)∣+∣x2(1)−x2(2)∣\rd_{12}=|x^{(1)}_1-x^{(2)}_1|+|x^{(1)}_2-x^{(2)}_2|\rd12=∣x1(1)−x1(2)∣+∣x2(1)−x2(2)∣\nn维平面上曼哈顿计算公式：\nd12=∑i=1n∣xi(1)−xi(2)∣\rd_{12}=\\sum\\limits_{i=1}^n|x^{(1)}_i-x^{(2)}_i|\rd12=i=1∑n∣xi(1)−xi(2)∣\n加权投票\nknn算法最后决定样本属于哪个类别，其实好比就是在投票，哪个类别票数多，则该样本属于哪个类别。而如果出现票数相同的情况，我们可以给每一票加上一个权重，用来表示每一票的重要性，这样就可以解决票数相同的问题了。很明显，距离越近的样本所投的一票应该越重要，此时我们可以将距离的倒数作为权重赋予每一票。\n\n如上图，虽然蓝色正方形与红色三角形数量一样，但是根据加权投票的规则，绿色的圆应该属于蓝色正方形这个类别。\n"},"Chapter4/k-近邻算法流程.html":{"url":"Chapter4/k-近邻算法流程.html","title":"4.3:k-近邻算法流程","keywords":"","body":"4.3:k-近邻算法流程\nknn算法不需要训练模型，只是根据离样本最近的几个样本类型来判别该样本类型，所以流程非常简单：\n\n1.计算出新样本与每一个样本的距离\n2.找出距离最近的k个样本\n3.根据加权投票规则得到新样本的类别\n\n"},"Chapter4/动手实现k-近邻.html":{"url":"Chapter4/动手实现k-近邻.html","title":"4.4:动手实现k-近邻","keywords":"","body":"4.4:动手实现k-近邻\nknn算法实现python代码如下：\n#encoding=utf8\nimport numpy as np\n\ndef knn_clf(k,train_feature,train_label,test_feature):\n    '''\n    input:\n        k(int):最近邻样本个数\n        train_feature(ndarray):训练样本特征\n        train_label(ndarray):训练样本标签\n        test_feature(ndarray):测试样本特征\n    output:\n        predict(ndarray):测试样本预测标签\n    '''\n    #初始化预测结果\n    predict = np.zeros(test_feature.shape[0],).astype('int')\n    #对测试集每一个样本进行遍历\n    for i in range(test_feature.shape[0]):\n        #测试集第i个样本到训练集每一个样本的距离\n        distance = np.sqrt(np.power(np.tile(test_feature[i],(train_feature.shape[0],1))-train_feature,2).sum(axis=1))\n        #最近的k个样本的距离\n        distance_k = np.sort(distance)[:k]\n        #最近的k个样本的索引\n        nearest = np.argsort(distance)[:k]\n        #最近的k个样本的标签\n        topK = [train_label[i] for i in nearest]\n        #初始化进行投票的字典，字典的键为标签，值为投票分数\n        votes = {}\n        #初始化最大票数\n        max_count = 0\n        #进行投票\n        for j,label in enumerate(topK):\n            #如果标签在字典的键中则投票计分\n            if label in votes.keys():\n                votes[label] += 1/(distance_k[j]+1e-10)#防止分母为0\n                #如果评分最高则将预测值更新为对应标签\n                if votes[label] > max_count:\n                    max_count = votes[label]\n                    predict[i] = label\n            #如果标签不在字典中则将标签加入字典的键，同时计入相应的分数\n            else:\n                votes[label] = 1/(distance_k[j]+1e-10)\n                if votes[label] > max_count:\n                    max_count = votes[label]\n                    predict[i] = label\n    return predict\n\n"},"Chapter4/实战案例.html":{"url":"Chapter4/实战案例.html","title":"4.5:实战案例","keywords":"","body":"4.5:实战案例\n手写数字数据\n手写数字数据集一共有1797个样本，每个样本有64个特征。每个特征的值为0-255之间的像素，我们的任务就是根据这64个特征值识别出该数字属于0-9十个类别中的哪一个。\n我们可以使用sklearn直接对数据进行加载，代码如下：\nfrom sklearn.datasets import load_digits\n#加载手写数字数据集\ndigits = load_digits()\n#获取数据特征与标签\nx,y = digits .data,digits .target\n\n当然，每一个样本就是一个数字，我们可以把它还原为8x8的大小进行查看：\nimport matplotlib.pyplot as plt\n\nimg = x[0].reshape(8,8)\nplt.imshow(img)\n\n\n然后我们划分出训练集与测试集，训练集用来训练模型，测试集用来检测模型性能。代码如下：\nfrom sklearn.model_selection import train_test_split\n#划分训练集测试集，其中测试集样本数为整个数据集的20%\ntrain_feature,test_feature,train_label,test_label = train_test_split(x,y,test_size=0.2,random_state=666)\n\n进行识别\n接下来就只需要调用之前实现的knn_clf方法就可以对测试集的手写数字进行识别了：\npredict = knn_clf(3,train_feature,train_label,test_feature)\npredict\n>>>array([1, 5, 0, 7, 1, 0, 6, 1, 5, 4, 9, 2, 7, 8, 4, 6, 9, 3, 7, 4, 7, 1,\n       8, 6, 0, 9, 6, 1, 3, 7, 5, 9, 8, 3, 2, 8, 8, 1, 1, 0, 7, 9, 0, 0,\n       8, 7, 2, 7, 4, 3, 4, 3, 4, 0, 4, 7, 0, 5, 5, 5, 2, 1, 7, 0, 5, 1,\n       8, 3, 3, 4, 0, 3, 7, 4, 3, 4, 2, 9, 7, 3, 2, 5, 3, 4, 1, 5, 5, 2,\n       9, 2, 2, 2, 2, 7, 0, 8, 1, 7, 4, 2, 3, 8, 2, 3, 3, 0, 2, 9, 9, 2,\n       3, 2, 8, 1, 1, 9, 1, 2, 0, 4, 8, 5, 4, 4, 7, 6, 7, 6, 6, 1, 7, 5,\n       6, 3, 8, 3, 7, 1, 8, 5, 3, 4, 7, 8, 5, 0, 6, 0, 6, 3, 7, 6, 5, 6,\n       2, 2, 2, 3, 0, 7, 6, 5, 6, 4, 1, 0, 6, 0, 6, 4, 0, 9, 3, 8, 1, 2,\n       3, 1, 9, 0, 7, 6, 2, 9, 3, 5, 3, 4, 6, 3, 3, 7, 4, 9, 2, 7, 6, 1,\n       6, 8, 4, 0, 3, 1, 0, 9, 9, 9, 0, 1, 8, 6, 8, 0, 9, 5, 9, 8, 2, 3,\n       5, 3, 0, 8, 7, 4, 0, 3, 3, 3, 6, 3, 3, 2, 9, 1, 6, 9, 0, 4, 2, 2,\n       7, 9, 1, 6, 7, 6, 3, 9, 1, 9, 3, 4, 0, 6, 4, 8, 5, 3, 6, 3, 1, 4,\n       0, 4, 4, 8, 7, 9, 1, 5, 2, 7, 0, 9, 0, 4, 4, 0, 1, 0, 6, 4, 2, 8,\n       5, 0, 2, 6, 0, 1, 8, 2, 0, 9, 5, 6, 2, 0, 5, 0, 9, 1, 4, 7, 1, 7,\n       0, 6, 6, 8, 0, 2, 2, 6, 9, 9, 7, 5, 1, 7, 6, 4, 6, 1, 9, 4, 7, 1,\n       3, 7, 8, 1, 6, 9, 8, 3, 2, 4, 8, 7, 5, 5, 6, 9, 9, 8, 5, 0, 0, 4,\n       9, 3, 0, 4, 9, 4, 2, 5])\n\n再根据测试集标签即真实分类结果，计算出正确率：\nacc = np.mean(predict==test_label)\nacc\n>>>0.994\n\n可以看到，使用knn对手写数字进行识别，正确率能达到99%以上。\n"},"Chapter5/":{"url":"Chapter5/","title":"第五章 线性回归","keywords":"","body":"第五章 线性回归\n"},"Chapter5/线性回归算法思想.html":{"url":"Chapter5/线性回归算法思想.html","title":"5.1:线性回归算法思想","keywords":"","body":"5.1:线性回归算法思想\n简单线性回归\n在生活中，我们常常能碰到这么一种情况，一个变量会跟着另一个变量的变化而变化，如圆的周长与半径的关系，当圆的半径确定了，那么周长也就确定了。还有一种情况就是，两个变量之间看似存在某种关系，但又没那么确定，如青少年的身高与体重，他们存在一种近似的线性关系：\n身高/cm = 体重/kg +105\n但是，并不是每个青少年都符合这个公式，只能说每个青少年的身高体重都存在这么一种近似的线性关系。这就是其实就是简单的线性回归，那么，到底什么是线性回归呢？假如我们将青少年的身高和体重值作为坐标，不同人的身高体重就会在平面上构成不同的坐标点，然后用一条直线，尽可能的去拟合这些点，这就是简单的线性回归。\n\n简单的线性回归模型如下：\ny=wx+b\ny = wx+b\ny=wx+b\n其中x表示特征值(如：体重值)，w表示权重，b表示偏置，y表示标签(如：身高值)。\n多元线性回归\n简单线性回归中，一个变量跟另一个变量的变化而变化，但是生活中，还有很多变量，可能由多个变量的变化决定着它的变化，比如房价，影响它的因素可能有：房屋面积、地理位置等等。如果我们要给它们建立出近似的线性关系，这就是多元线性回归，多元线性回归模型如下：\ny=b+w1x1+w2x2+...+wnxn\ny=b+w_1x_1+w_2x_2+...+w_nx_n\ny=b+w1x1+w2x2+...+wnxn\n其中，xix_ixi表示第i个特征，wiw_iwi表示第i个特征对于的权重，b表示偏置，y表示标签。\n"},"Chapter5/线性回归算法原理.html":{"url":"Chapter5/线性回归算法原理.html","title":"5.2:线性回归算法原理","keywords":"","body":"5.2:线性回归算法原理\n线性回归训练流程\n我们已经知道线性回归模型如下：\ny=b+w1x1+w2x2+...+wnxn\ny = b +w_1x_1+w_2x_2+...+w_nx_n\ny=b+w1x1+w2x2+...+wnxn\n为了方便，我们稍微将模型进行变换：\ny=w0x0+w1x1+w2x2+...+wnxn\ny = w_0x_0 +w_1x_1+w_2x_2+...+w_nx_n\ny=w0x0+w1x1+w2x2+...+wnxn\n其中x0=1,w0=b,通过向量化公式可写成如下形式：\nY=X.W\nY=X.W\nY=X.W\nW=(w0,w1,...,wn)\nW = (w_0,w_1,...,w_n)\nW=(w0,w1,...,wn)\nX=(1,x1,...,xn)\nX = (1,x_1,...,x_n)\nX=(1,x1,...,xn)\n而我们的目的就是找出能够正确预测的多元线性回归模型，即找出正确的W（即权重与偏置）。那么如何寻找呢？通常在监督学习里面都会使用这么一个套路，构造一个损失函数，用来衡量真实值与预测值之间的差异，然后将问题转化为最优化损失函数。既然损失函数是用来衡量真实值与预测值之间的差异那么很多人自然而然的想到了用所有真实值与预测值的差的绝对值来表示损失函数。不过带绝对值的函数不容易求导，所以采用MSE(均方误差)作为损失函数，公式如下：\nloss=1m∑i=1m(y(i)−p(i))2\nloss = \\frac{1}{m}\\sum\\limits_{i=1}^m(y^{(i)}-p^{(i)})^2\nloss=m1i=1∑m(y(i)−p(i))2\n其中p表示预测值，y表示真实值，m为样本总个数，i表示第i个样本。最后，我们再使用正规方程解来求得我们所需要的参数。\n线性回归模型训练流程图如下：\n\n正规方程解\n对线性回归模型，假设训练集中m个训练样本，每个训练样本中有n个特征，可以使用矩阵的表示方法，预测函数可以写为：\nY=X.W\nY = X.W\nY=X.W\n其损失函数可以表示为\nloss=1m(Y−X.W)T(Y−X.W)\nloss = \\frac{1}{m}(Y-X.W)^T(Y-X.W)\nloss=m1(Y−X.W)T(Y−X.W)\n其中，标签Y为m行1列的矩阵，训练特征X为m行(n+1)列的矩阵，回归系数W为(n+1)行1列的矩阵，对W求导，并令其导数为零可解得：\nW=(XTX)−1XTY\nW=(X^TX)^{-1}X^TY\nW=(XTX)−1XTY\n这个就是正规方程解，我们可以通过正规方程解直接求得我们所需要的参数。\n"},"Chapter5/线性回归算法流程.html":{"url":"Chapter5/线性回归算法流程.html","title":"5.3:线性回归算法流程","keywords":"","body":"5.3:线性回归算法流程\n我们最终的目的是通过训练出来的线性回归模型对测试集数据进行预测，算法实现流程如下：\n\n1.将x0=1加入训练数据\n2.使用正规方程解求得参数\n3.将x0=1加入测试数据\n4.对测试集数据进行预测\n\n"},"Chapter5/动手实现线性回归.html":{"url":"Chapter5/动手实现线性回归.html","title":"5.4:动手实现线性回归","keywords":"","body":"5.4:动手实现线性回归\n线性回归python实现代码如下：\n#encoding=utf8\nimport numpy as np\n\ndef lr(train_feature,train_label,test_feature):\n    '''\n    input:\n        train_feature(ndarray):训练样本特征\n        train_label(ndarray):训练样本标签\n        test_feature(ndarray):测试样本特征\n    output:\n        predict(ndarray):测试样本预测标签\n    '''\n    #将x0=1加入训练数据\n    train_x = np.hstack([np.ones((len(train_feature),1)),train_feature])\n    #使用正规方程解求得参数\n    theta =np.linalg.inv(train_x.T.dot(train_x)).dot(train_x.T).dot(train_label)\n    #将x0=1加入测试数据\n    test_x = np.hstack([np.ones((len(test_feature),1)),test_feature])\n    #求得测试集预测标签    \n    predict = test_x.dot(theta)\n    return predict\n\n"},"Chapter5/实战案例.html":{"url":"Chapter5/实战案例.html","title":"5.5:实战案例","keywords":"","body":"5.5:实战案例\n波士顿房价数据\n波士顿房价数据集共有506条房价数据，每条数据包括对指定房屋的13项数值型特征和目标房价组成。我们需要通过数据特征来对目标房价进行预测。\n数据集中部分数据与标签如下图所示：\n\n\nsklearn中已经提供了波士顿房价数据集的相关接口，想要使用该数据集可以使用如下代码：\nfrom sklearn import datasets\n#加载波士顿房价数据集\nboston = datasets.load_boston()\n#X表示特征，y表示目标房价\nx = boston.data\ny = boston.target\n\n然后再对数据集进行划分：\nfrom sklearn.model_selection import train_test_split\n#划分训练集测试集，所有样本的20%作为测试集\ntrain_feature,test_feature,train_label,test_label = train_test_split(x,y,test_size=0.2,random_state=666)\n\n进行预测\n同样的只需要调用之前实现线性回归方法就可以对测试集的波士顿房价数据进行预测了：\npredict = lr(train_feature,train_label,test_feature)\n>>>predict\narray([27.14328365, 23.03653632, 27.00098113, 34.67246356, 22.9249281 ,\n       21.27666411, 15.67682012, 23.71041177, 24.9170328 , 18.94485146,\n        4.21475157, 24.91145159, 20.98995302, 18.43508891, 24.17666486,\n       26.84239278, 27.83397467, 13.52699359, 18.45498398, 28.42388411,\n       30.59256907, 13.41724252,  8.12085396, 35.51572129, 25.67615918,\n       17.16601994, 20.37433719, 13.09756854, 34.29369038, 23.73452722,\n       39.80575322,  8.23996654, 24.79976309, 17.93534789, 23.166615  ,\n       19.77561659, 35.15958711, 35.62614752, 21.48402467, 13.53651885,\n       23.8764859 , 22.76090085, 27.69433621, 18.25312903, 28.24166439,\n       11.37889658, 27.10532052, 32.76787747, 29.42762069, 24.90135914,\n       27.29432351, 33.19296658, 26.14048342, 23.62626694, 27.59078519,\n       20.00241919, 14.46427082, 20.0119397 , 19.81015781, 13.93309224,\n       20.96227953, 25.93383085, 30.17587814, 18.06438076, 12.03215906,\n       11.3801673 , 26.81093528, 22.56148123, 22.95599483, 25.79865129,\n       10.10532755, 33.63114297, 17.81932257, 17.21896388, 39.33351986,\n       14.91994896, 18.19524145, 24.94373123, 20.09101825, 31.48389087,\n       32.8430831 , 23.95919903,  9.77345135, 31.55307878, 30.55370904,\n       23.20332797, 21.90050123, 13.5557125 , 18.27957707, 25.0240593 ,\n       19.54159097, 36.39430746, 24.02473259, 33.08973723, 21.71311184,\n       17.37919862, 26.67885309, 27.42896672, 13.1943355 ,  0.57642556,\n       19.69396665, 14.18869608])\n\n衡量线性回归的性能指标\n对于分类问题，我们可以使用正确率来衡量模型的性能好坏，很明显，回归问题并不能使用正确率来衡量，那么，回归问题可以使用哪些指标用来评估呢？\nMSE\nMSE （Mean Squared Error）叫做均方误差,公式如下：\nmse=1m∑i=1m(y(i)−p(i))2\nmse = \\frac{1}{m}\\sum\\limits_{i=1}^m(y^{(i)}-p^{(i)})^2\nmse=m1i=1∑m(y(i)−p(i))2\n其中yi表示第i个样本的真实标签，pi表示第i个样本的预测标签。线性回归的目的就是让损失函数最小。那么，模型训练出来了，我们再测试集上用损失函数来评估也是可以的。\nRMSE\nRMSE（Root Mean Squard Error）均方根误差，公式如下：\nrmse=1m∑i=1m(y(i)−p(i))2\nrmse = \\sqrt{\\frac{1}{m}\\sum\\limits_{i=1}^m(y^{(i)}-p^{(i)})^2}\nrmse=⎷m1i=1∑m(y(i)−p(i))2\nRMSE其实就是MSE开个根号。有什么意义呢？其实实质是一样的。只不过用于数据更好的描述。\n例如：要做房价预测，每平方是万元，我们预测结果也是万元。那么差值的平方单位应该是千万级别的。那我们不太好描述自己做的模型效果。怎么说呢？我们的模型误差是多少千万？于是干脆就开个根号就好了。我们误差的结果就跟我们数据是一个级别的了，在描述模型的时候就说，我们模型的误差是多少万元。\nMAE\nMAE(平均绝对误差)，公式如下：\nmae=1m∑i=1m∣y(i)−p(i)∣\nmae = \\frac{1}{m}\\sum\\limits_{i=1}^m|y^{(i)}-p^{(i)}|\nmae=m1i=1∑m∣y(i)−p(i)∣\nMAE虽然不作为损失函数，确是一个非常直观的评估指标，它表示每个样本的预测标签值与真实标签值的L1距离。\nR-Squared\n上面的几种衡量标准针对不同的模型会有不同的值。比如说预测房价 那么误差单位就是万元。数子可能是3，4，5之类的。那么预测身高就可能是0.1，0.6之类的。没有什么可读性，到底多少才算好呢？不知道，那要根据模型的应用场景来。 看看分类算法的衡量标准就是正确率，而正确率又在0～1之间，最高百分之百。最低0。那么线性回归有没有这样的衡量标准呢？R-Squared就是这么一个指标，公式如下：\nR2=1−∑i=1m(p(i)−y(i))2∑i=1m(ymean(i)−y(i))2\nR^2=1-\\frac{\\sum\\limits_{i=1}^m(p^{(i)}-y^{(i)})^2}{\\sum\\limits_{i=1}^m(y_{mean}^{(i)}-y^{(i)})^2}\nR2=1−i=1∑m(ymean(i)−y(i))2i=1∑m(p(i)−y(i))2\n为什么这个指标会有刚刚我们提到的性能呢？我们分析下公式：\n\n其实分子表示的是模型预测时产生的误差，分母表示的是对任意样本都预测为所有标签均值时产生的误差，由此可知：\n\n1.当我们的模型不犯任何错误时，取最大值1。\n2.当我们的模型性能跟基模型性能相同时，取0。\n3.如果为负数，则说明我们训练出来的模型还不如基准模型，此时，很有可能我们的数据不存在任何线性关系。\n\n其中，基准模型值的随机瞎猜的模型。\n这里使用python实现了MSE，R-Squared方法，代码如下：\nimport numpy as np\n\n#mse\ndef mse_score(y_predict,y_test):\n    mse = np.mean((y_predict-y_test)**2)\n    return mse\n#r2\ndef r2_score(y_predict,y_test):\n    '''\n    input:y_predict(ndarray):预测值\n          y_test(ndarray):真实值\n    output:r2(float):r2值\n    '''\n    r2 = 1 - mse_score(y_predict,y_test)/np.var(y_test)\n    return r2\n我们可以根据求得的预测值，计算出MSE值与R-Squared值:\nmse = mse_score(predict,test_label)\nmse\n>>>27.22\nr2 = r2_score(predict,test_label)\nr2\n>>>0.63\n\n"},"Chapter6/":{"url":"Chapter6/","title":"第六章 决策树","keywords":"","body":"第六章 决策树\n"},"Chapter6/决策树算法思想.html":{"url":"Chapter6/决策树算法思想.html","title":"6.1:决策树算法思想","keywords":"","body":"6.1:决策树算法思想\n决策树是一种可以用于分类与回归的机器学习算法，但主要用于分类。用于分类的决策树是一种描述对实例进行分类的树形结构。决策树由结点和边组成，其中结点分为内部结点和叶子结点，内部结点表示一个特征或者属性，叶子结点表示标签。\n决策树说通俗点就是一棵能够替我们做决策的树，或者说是我们人类在要做决策时脑回路的一种表现形式，我们可以从下面这个例子来了解决策树是什么。\n在炎热的夏天，没有什么比冰镇后的西瓜更能令人感到心旷神怡的了。现在我要去水果店买西瓜，但怎样我才会买这个西瓜呢？那么，有可能我会有以下这个决策过程：\n\n假设现在水果店里有3个西瓜，它们的属性如下：\n\n\n\n编号\n瓤是否够红\n够不够冰\n是否便宜\n是否有籽\n\n\n\n\n1\n是\n否\n是\n否\n\n\n2\n是\n是\n否\n是\n\n\n3\n否\n是\n是\n否\n\n\n\n那么根据我的决策过程我会买1和2号西瓜。这个帮助我选择西瓜的过程，就是一个决策树。由之前介绍的知识可以知道，黄色部分为内部节点，蓝色部分为叶子节点。\n"},"Chapter6/决策树算法原理.html":{"url":"Chapter6/决策树算法原理.html","title":"6.2:决策树算法原理","keywords":"","body":"6.2:决策树算法原理\n我们已经知道，构造一棵决策树其实就是根据数据的特征(内部节点)对数据一步一步的进行划分，从而达到分类的目的。但是，每一步根据哪个特征来进行划分呢？这个就是构造决策树的关键。其实构造决策树时会遵循一个指标，有的是按照信息增益来构建，如ID3算法；有的是信息增益率来构建，如C4.5算法；有的是按照基尼系数来构建的，如CART算法。但不管是使用哪种构建算法，决策树的构建过程通常都是一个递归选择最优特征，并根据特征对训练集进行分割，使得对各个子数据集有一个最好的分类的过程。这里我们以ID3算法为例，详细介绍构建决策树相关知识。\n信息熵\n信息是个很抽象的概念。人们常常说信息很多，或者信息较少，但却很难说清楚信息到底有多少。比如一本五十万字的中文书到底有多少信息量。\n直到1948年，香农提出了“信息熵”的概念，才解决了对信息的量化度量问题。信息熵这个词是香农从热力学中借用过来的。热力学中的热熵是表示分子状态混乱程度的物理量。香农用信息熵的概念来描述信源的不确定度。信源的不确定性越大，信息熵也越大。\n从机器学习的角度来看，信息熵表示的是信息量的期望值。如果数据集中的数据需要被分成多个类别，则信息量 I(xi)I(x_i)I(xi)的定义如下：\n其中xix_ixi表示多个类别中的第i个类别，p(xi)p(x_i)p(xi)表示概率：\nI(Xi)=−log2p(xi)\nI(X_i)=-log_2p(x_i)\nI(Xi)=−log2p(xi)\n由于信息熵是信息量的期望值，所以信息熵H(X)H(X)H(X)的定义如下(其中n为数据集中类别的数量)：\nH(X)=−∑i=1np(xi)log2p(xi)\nH(X)=-\\sum\\limits_{i=1}^np(x_i)log_2p(x_i)\nH(X)=−i=1∑np(xi)log2p(xi)\n从这个公式也可以看出，如果概率是0或者是1的时候，熵就是0。（因为这种情况下随机变量的不确定性是最低的），那如果概率是0.5也就是五五开的时候，此时熵达到最大，也就是1。（就像扔硬币，你永远都猜不透你下次扔到的是正面还是反面，所以它的不确定性非常高）。所以呢，熵越大，不确定性就越高。\n条件熵\n在实际的场景中，我们可能需要研究数据集中某个特征等于某个值时的信息熵等于多少，这个时候就需要用到条件熵。条件熵H(Y|X)表示特征X为某个值的条件下，类别为Y的熵。条件熵的计算公式如下：\nH(Y∣X)=∑i=1npiH(Y∣X=xi)\nH(Y|X)=\\sum\\limits_{i=1}^np_iH(Y|X=x_i)\nH(Y∣X)=i=1∑npiH(Y∣X=xi)\n信息增益\n现在已经知道了什么是熵，什么是条件熵。接下来就可以看看什么是信息增益了。所谓的信息增益就是表示我已知条件X后能得到信息Y的不确定性的减少程度。\n就好比，我在玩读心术。你心里想一件东西，我来猜。我已开始什么都没问你，我要猜的话，肯定是瞎猜。这个时候我的熵就非常高。然后我接下来我会去试着问你是非题，当我问了是非题之后，我就能减小猜测你心中想到的东西的范围，这样其实就是减小了我的熵。那么我熵的减小程度就是我的信息增益。\n所以信息增益如果套上机器学习的话就是，如果把特征A对训练集D的信息增益记为g(D, A)的话，那么g(D, A)的计算公式就是：\ng(D,A)=H(D)−H(D,A)\ng(D,A)=H(D)-H(D,A)\ng(D,A)=H(D)−H(D,A)\n为了更好的解释熵，条件熵，信息增益的计算过程，下面通过示例来描述。假设我现在有这一个数据集，第一列是编号，第二列是性别，第三列是活跃度，第四列是客户是否流失的标签（0:表示未流失，1:表示流失）。\n\n\n\n编号\n性别\n活跃度\n是否流失\n\n\n\n\n1\n男\n高\n0\n\n\n2\n女\n中\n0\n\n\n3\n男\n低\n1\n\n\n4\n女\n高\n0\n\n\n5\n男\n高\n0\n\n\n6\n男\n中\n0\n\n\n7\n男\n中\n1\n\n\n8\n女\n中\n0\n\n\n9\n女\n低\n1\n\n\n10\n女\n中\n0\n\n\n11\n女\n高\n0\n\n\n12\n男\n低\n1\n\n\n13\n女\n低\n1\n\n\n14\n男\n高\n0\n\n\n15\n男\n高\n0\n\n\n\n假如要算性别和活跃度这两个特征的信息增益的话，首先要先算总的熵和条件熵。总的熵其实非常好算，就是把标签作为随机变量X。上表中标签只有两种（0和1）因此随机变量X的取值只有0或者1。所以要计算熵就需要先分别计算标签为0的概率和标签为1的概率。从表中能看出标签为0的数据有10条，所以标签为0的概率等于2/3。标签为1的概率为1/3。所以熵为：\n−13∗log(13)−23∗log(23)=0.9182\n-\\frac{1}{3}*log(\\frac{1}{3})-\\frac{2}{3}*log(\\frac{2}{3}) = 0.9182\n−31∗log(31)−32∗log(32)=0.9182\n接下来就是条件熵的计算，以性别为男的熵为例。表格中性别为男的数据有8条，这8条数据中有3条数据的标签为1，有5条数据的标签为0。所以根据条件熵的计算公式能够得出该条件熵为：\n−38∗log(38)−58∗log(58)=0.9543\n-\\frac{3}{8}*log(\\frac{3}{8})-\\frac{5}{8}*log(\\frac{5}{8}) = 0.9543\n−83∗log(83)−85∗log(85)=0.9543\n根据上述的计算方法可知，总熵为：\n−515∗log(515)−1015∗log(1015)=0.9182\n-\\frac{5}{15}*log(\\frac{5}{15})-\\frac{10}{15}*log(\\frac{10}{15}) = 0.9182\n−155∗log(155)−1510∗log(1510)=0.9182\n性别为男的熵为：\n−38∗log(38)−58∗log(58)=0.9543\n-\\frac{3}{8}*log(\\frac{3}{8})-\\frac{5}{8}*log(\\frac{5}{8}) = 0.9543\n−83∗log(83)−85∗log(85)=0.9543\n性别为女的熵为：\n−27∗log(27)−57∗log(57)=0.8631\n-\\frac{2}{7}*log(\\frac{2}{7})-\\frac{5}{7}*log(\\frac{5}{7}) = 0.8631\n−72∗log(72)−75∗log(75)=0.8631\n活跃度为低的熵为：\n−44∗log(44)=0\n-\\frac{4}{4}*log(\\frac{4}{4}) = 0\n−44∗log(44)=0\n活跃度为中的熵为：\n−15∗log(15)−45∗log(45)=0.7219\n-\\frac{1}{5}*log(\\frac{1}{5})-\\frac{4}{5}*log(\\frac{4}{5}) = 0.7219\n−51∗log(51)−54∗log(54)=0.7219\n活跃度为高的熵为：\n−0−66∗log(66)=0\n-0-\\frac{6}{6}*log(\\frac{6}{6}) = 0\n−0−66∗log(66)=0\n现在有了总的熵和条件熵之后就能算出性别和活跃度这两个特征的信息增益了。\n性别的信息增益=总的熵-(8/15)性别为男的熵-(7/15)性别为女的熵=0.0064\n活跃度的信息增益=总的熵-(6/15)活跃度为高的熵-(5/15)活跃度为中的熵-(4/15)活跃度为低的熵=0.6776 \n那信息增益算出来之后有什么意义呢？回到读心术的问题，为了我能更加准确的猜出你心中所想，我肯定是问的问题越好就能猜得越准！换句话来说我肯定是要想出一个信息增益最大（减少不确定性程度最高）的问题来问你。其实ID3算法也是这么想的。ID3算法的思想是从训练集D中计算每个特征的信息增益，然后看哪个最大就选哪个作为当前结点。然后继续重复刚刚的步骤来构建决策树。\n"},"Chapter6/决策树算法流程.html":{"url":"Chapter6/决策树算法流程.html","title":"6.3:决策树算法流程","keywords":"","body":"6.3:决策树算法流程\n我们最终的目的是根据创建的决策树模型对测试集数据进行预测，算法实现流程如下：\n\n1.计算训练样本信息增益\n2.获得信息增益最高的特征\n3.递归创建决策树\n4.根据决策树模型对测试集数据进行预测\n\n"},"Chapter6/动手实现决策树.html":{"url":"Chapter6/动手实现决策树.html","title":"6.4:动手实现决策树","keywords":"","body":"6.4:动手实现决策树\n#encoding=utf8\nimport numpy as np\n\n# 计算熵\ndef calcInfoEntropy(label):\n    '''\n    label(narray):样本标签\n    '''\n    label_set = set(label)\n    result = 0\n    for l in label_set:\n        count = 0\n        for j in range(len(label)):\n            if label[j] == l:\n                count += 1\n        # 计算标签在数据集中出现的概率\n        p = count / len(label)\n        # 计算熵\n        result -= p * np.log2(p)\n    return result\n\n#计算条件熵\ndef calcHDA(feature,label,index,value):\n    '''\n    input:\n        feature(ndarray):样本特征\n        label(ndarray):样本标签\n        index(int):需要使用的特征列索引\n        value(int):index所表示的特征列中需要考察的特征值\n    output:\n        HDA(float):信息熵\n    '''\n    count = 0\n    # sub_feature和sub_label表示根据特征列和特征值分割出的子数据集中的特征和标签\n    sub_feature = []\n    sub_label = []\n    for i in range(len(feature)):\n        if feature[i][index] == value:\n            count += 1\n            sub_feature.append(feature[i])\n            sub_label.append(label[i])\n    pHA = count / len(feature)\n    e = calcInfoEntropy(sub_label)\n    HDA = pHA * e\n    return HDA\n\n#计算信息增益\ndef calcInfoGain(feature, label, index):\n    '''\n    input:\n        feature(ndarry):测试用例中字典里的feature\n        label(ndarray):测试用例中字典里的label\n        index(int):测试用例中字典里的index，即feature部分特征列的索引。该索引指的是feature中第几个特征，如index:0表示使用第一个特征来计算信息增益。\n    output:\n        InfoGain(float):信息增益\n    '''\n    base_e = calcInfoEntropy(label)\n    f = np.array(feature)\n    # 得到指定特征列的值的集合\n    f_set = set(f[:, index])\n    sum_HDA = 0\n    # 计算条件熵\n    for value in f_set:\n        sum_HDA += calcHDA(feature, label, index, value)\n    # 计算信息增益\n    InfoGain = base_e - sum_HDA\n    return InfoGain\n\n# 获得信息增益最高的特征\ndef getBestFeature(feature, label):\n    '''\n    input:\n        feature(ndarray):样本特征\n        label(ndarray):样本标签\n    output:\n        best_feature(int):信息增益最高的特征\n    '''\n    max_infogain = 0\n    best_feature = 0\n    for i in range(len(feature[0])):\n        infogain = calcInfoGain(feature, label, i)\n        if infogain > max_infogain:\n            max_infogain = infogain\n            best_feature = i\n    return best_feature\n\n#创建决策树\ndef createTree(feature, label):\n    '''\n    input:\n        feature(ndarray):训练样本特征\n        label(ndarray):训练样本标签\n    output:\n        tree(dict):决策树模型    \n    '''\n    # 样本里都是同一个label没必要继续分叉了\n    if len(set(label)) == 1:\n        return label[0]\n    # 样本中只有一个特征或者所有样本的特征都一样的话就看哪个label的票数高\n    if len(feature[0]) == 1 or len(np.unique(feature, axis=0)) == 1:\n        vote = {}\n        for l in label:\n            if l in vote.keys():\n                vote[l] += 1\n            else:\n                vote[l] = 1\n        max_count = 0\n        vote_label = None\n        for k, v in vote.items():\n            if v > max_count:\n                max_count = v\n                vote_label = k\n        return vote_label\n    # 根据信息增益拿到特征的索引\n    best_feature = getBestFeature(feature, label)\n    tree = {best_feature: {}}\n    f = np.array(feature)\n    # 拿到bestfeature的所有特征值\n    f_set = set(f[:, best_feature])\n    # 构建对应特征值的子样本集sub_feature, sub_label\n    for v in f_set:\n        sub_feature = []\n        sub_label = []\n        for i in range(len(feature)):\n            if feature[i][best_feature] == v:\n                sub_feature.append(feature[i])\n                sub_label.append(label[i])\n        # 递归构建决策树\n        tree[best_feature][v] = createTree(sub_feature, sub_label)\n    return tree\n\n#决策树分类\ndef dt_clf(train_feature,train_label,test_feature):\n    '''\n    input:\n        train_feature(ndarray):训练样本特征\n        train_label(ndarray):训练样本标签\n        test_feature(ndarray):测试样本特征\n    output:\n        predict(ndarray):测试样本预测标签     \n    '''\n    #创建决策树\n    tree = createTree(train_feature,train_label)\n    result = []\n    #根据tree与特征进行分类\n    def classify(tree,test_feature):\n        #如果tree是叶子节点，返回tree\n        if not isinstance(tree,dict):\n            return tree\n        #根据特征值走入tree中的分支\n        t_index,t_value = list(tree.items())[0]\n        f_value = test_feature[t_index]\n        #如果分支依然是tree\n        if isinstance(t_value,dict):\n            #根据tree与特征进行分类\n            classLabel = classify(tree[t_index][f_value],test_feature)\n            return classLabel\n        else:\n            #返回特征值\n            return t_value\n    for f in test_feature:\n        result.append(classify(tree,f))\n    predict = np.array(result)\n    return predict\n\n"},"Chapter6/实战案例.html":{"url":"Chapter6/实战案例.html","title":"6.5:实战案例","keywords":"","body":"6.5:实战案例\n鸢尾花数据\n鸢尾花数据集是一类多重变量分析的数据集，一共有150个样本，通过花萼长度，花萼宽度，花瓣长度，花瓣宽度 4个特征预测鸢尾花卉属于（Setosa，Versicolour，Virginica）三个种类中的哪一类。\n数据集中部分数据如下所示：\n\n\n\n花萼长度\n花萼宽度\n花瓣长度\n花瓣宽度\n\n\n\n\n5.1\n3.5\n1.4\n0.2\n\n\n4.9\n3.2\n1.4\n0.2\n\n\n4.7\n3.1\n1.3\n0.2\n\n\n\n其中每一行代表一个鸢尾花样本各个属性的值。\n数据集中部分标签如下图所示：\n\n\n\n标签\n\n\n\n\n0\n\n\n1\n\n\n2\n\n\n\n标签中的值0,1,2分别代表鸢尾花三种不同的类别。\n我们可以直接使用sklearn直接对数据进行加载，代码如下：\nfrom sklearn.datasets import load_iris\n#加载鸢尾花数据集\niris = load_iris()\n#获取数据特征与标签\nx,y = iris.data.astype(int),iris.target\n\n然后我们划分出训练集与测试集，训练集用来训练模型，测试集用来检测模型性能。代码如下：\nfrom sklearn.model_selection import train_test_split\n#划分训练集测试集，其中测试集样本数为整个数据集的20%\ntrain_feature,test_feature,train_label,test_label = train_test_split(x,y,test_size=0.2,random_state=666)\n\n进行分类\n然后我们再使用实现的决策树分类方法就可以对测试集数据进行分类：\npredict = dt_clf(train_feature,train_label,test_feature)\npredict\n>>>array([1, 2, 1, 2, 0, 1, 1, 2, 1, 1, 1, 0, 0, 0, 2, 1, 0, 2, 2, 2, 1, 0,2, 0, 1, 1, 0, 1, 2, 2])\n\n再根据测试集标签，可以计算出正确率：\nacc = np.mean(predict==test_label)\nacc\n>>>1.0\n\n可以看到，使用决策树对鸢尾花进行分类，正确率可以达到100%\n"},"Chapter7/":{"url":"Chapter7/","title":"第七章 k-均值","keywords":"","body":"第七章 k-均值\n"},"Chapter7/k-均值算法思想.html":{"url":"Chapter7/k-均值算法思想.html","title":"7.1:k-均值算法思想","keywords":"","body":"7.1:k-均值算法思想\nk-means是属于机器学习里面的非监督学习，通常是大家接触到的第一个聚类算法，其思想非常简单，是一种典型的基于距离的聚类算法。k-means（K-均值）聚类，之所以称为 K-均值 是因为它可以发现K个簇，且每个簇的中心采用簇中所含值的均值计算而成。簇内的样本连接紧密，而簇之间的距离尽量大。简单来讲，其思想就是物以类聚。\n"},"Chapter7/k-均值算法原理.html":{"url":"Chapter7/k-均值算法原理.html","title":"7.2:k-均值算法原理","keywords":"","body":"7.2:k-均值算法原理\n假设我们有k个簇：(c1,c2,...,ck)(c_1,c_2,...,c_k)(c1,c2,...,ck)\n则我们的目的就是使的簇内的每个点到簇的质心的距离最小，即最小化平方误差MSE：\n∑i=1k∑x∈ci(x−ui)2\n\\sum\\limits_{i=1}^k\\sum\\limits_{x\\in c_i}(x-u_i)^2\ni=1∑kx∈ci∑(x−ui)2\n其中，uiu_iui为质心，表达式为：\n1∣ci∣∑x∈cix\n\\frac{1}{|c_i|}\\sum\\limits_{x\\in c_i}x\n∣ci∣1x∈ci∑x\n∣ci∣|c_i|∣ci∣表示集合内样本个数。\n想要直接求得最小值是非常困难的，通常我们使用启发式的迭代方法，过程如下图：\n\n\n图b:假设k=2，我们最开始先随机初始2个质心(红色与蓝色的点)。\n图c:计算每个样本到两个质心的距离，并将其归为与其距离最近的质心那个簇。\n图d:更新质心，我们可以看到，红色与蓝色的点位置有了变化。\n图e:重新计算样本到质心距离，并重新划分样本属于哪个簇。\n图f:直到质心位置变换小于阈值，停止迭代。\n\n"},"Chapter7/k-均值算法流程.html":{"url":"Chapter7/k-均值算法流程.html","title":"7.3:k-均值算法流程","keywords":"","body":"7.3:k-均值算法流程\nk-means算法流程如下：\n\n1.随机初始k个点，作为类别中心。\n2.对每个样本将其标记为距离类别中心最近的类别。\n3.将每个类别的质心更新为新的类别中心。\n4.重复步骤2、3，直到类别中心的变换小于阈值。\n\n"},"Chapter7/动手实现k-均值.html":{"url":"Chapter7/动手实现k-均值.html","title":"7.4:动手实现k-均值","keywords":"","body":"7.4:动手实现k-均值\n#encoding=utf8\nimport numpy as np\n\n# 计算一个样本与数据集中所有样本的欧氏距离的平方\ndef euclidean_distance(one_sample, X):\n    '''\n    input:\n        one_sample(ndarray):单个样本\n        X(ndarray):所有样本\n    output:\n        distances(ndarray):单个样本到所有样本的欧氏距离平方\n    '''\n    one_sample = one_sample.reshape(1, -1)\n    distances = np.power(np.tile(one_sample, (X.shape[0], 1)) - X, 2).sum(axis=1)\n    return distances\n\n# 从所有样本中随机选取k个样本作为初始的聚类中心\ndef init_random_centroids(k,X):\n    '''\n    input:\n        k(int):聚类簇的个数\n        X(ndarray):所有样本\n    output:\n        centroids(ndarray):k个簇的聚类中心\n    '''\n    n_samples, n_features = np.shape(X)\n    centroids = np.zeros((k, n_features))\n    for i in range(k):\n        centroid = X[np.random.choice(range(n_samples))]\n        centroids[i] = centroid\n    return centroids\n\n# 返回距离该样本最近的一个中心索引[0, k)\ndef _closest_centroid(sample, centroids):\n    '''\n    input:\n        sample(ndarray):单个样本\n        centroids(ndarray):k个簇的聚类中心\n    output:\n        closest_i(int):最近中心的索引\n    '''\n    distances = euclidean_distance(sample, centroids)\n    closest_i = np.argmin(distances)\n    return closest_i\n\n# 将所有样本进行归类，归类规则就是将该样本归类到与其最近的中心\ndef create_clusters(k,centroids, X):\n    '''\n    input:\n        k(int):聚类簇的个数\n        centroids(ndarray):k个簇的聚类中心\n        X(ndarray):所有样本\n    output:\n        clusters(list):列表中有k个元素，每个元素保存相同簇的样本的索引\n    '''\n    clusters = [[] for _ in range(k)]\n    for sample_i, sample in enumerate(X):\n        centroid_i = _closest_centroid(sample, centroids)\n        clusters[centroid_i].append(sample_i)\n    return clusters\n\n# 对中心进行更新\ndef update_centroids(k,clusters, X):\n    '''\n    input:\n        k(int):聚类簇的个数\n        X(ndarray):所有样本\n    output:\n        centroids(ndarray):k个簇的聚类中心\n    '''\n    n_features = np.shape(X)[1]\n    centroids = np.zeros((k, n_features))\n    for i, cluster in enumerate(clusters):\n        centroid = np.mean(X[cluster], axis=0)\n        centroids[i] = centroid\n    return centroids\n\n# 将所有样本进行归类，其所在的类别的索引就是其类别标签\ndef get_cluster_labels(clusters, X):\n    '''\n    input:\n        clusters(list):列表中有k个元素，每个元素保存相同簇的样本的索引\n        X(ndarray):所有样本\n    output:\n        y_pred(ndarray):所有样本的类别标签\n    '''\n    y_pred = np.zeros(np.shape(X)[0])\n    for cluster_i, cluster in enumerate(clusters):\n        for sample_i in cluster:\n            y_pred[sample_i] = cluster_i\n    return y_pred\n\n# 对整个数据集X进行Kmeans聚类，返回其聚类的标签\ndef predict(k,X,max_iterations,varepsilon):\n    '''\n    input:\n        k(int):聚类簇的个数\n        X(ndarray):所有样本\n        max_iterations(int):最大训练轮数\n        varepsilon(float):最小误差阈值\n    output:\n        y_pred(ndarray):所有样本的类别标签\n    '''\n    # 从所有样本中随机选取k样本作为初始的聚类中心\n    centroids = init_random_centroids(k,X)\n    # 迭代，直到算法收敛(上一次的聚类中心和这一次的聚类中心几乎重合)或者达到最大迭代次数\n    for _ in range(max_iterations):\n        # 将所有进行归类，归类规则就是将该样本归类到与其最近的中心\n        clusters = create_clusters(k,centroids, X)\n        former_centroids = centroids\n        # 计算新的聚类中心\n        centroids = update_centroids(k,clusters, X)\n        # 如果聚类中心几乎没有变化，说明算法已经收敛，退出迭代\n        diff = centroids - former_centroids\n        if diff.any() \n"},"Chapter7/实战案例.html":{"url":"Chapter7/实战案例.html","title":"7.5:实战案例","keywords":"","body":"7.5:实战案例\n鸢尾花数据\n本次我们使用的仍然是鸢尾花数据，不过为了能够进行可视化我们只使用数据中的两个特征：\nfrom sklearn.datasets import load_iris\n\niris = load_iris()\nx,y = iris.data,iris.target\nx = x[:,2:]\n\n可视化数据分布：\nimport matplotlib.pyplot as plt\n\nplt.scatter(x[:,0],x[:,1])\nplt.show()\n\n可视化结果：\n\n我们可以先根据数据的真实标签查看数据类别情况：\nimport matplotlib.pyplot as plt\n\nplt.scatter(x[:,0],x[:,1],c=y)\nplt.show()\n\n效果如下：\n\n进行聚类\n最后，使用我们实现的k-means方法对数据进行聚类并查看聚类效果：\npredict = predict(3,x,500,0.0001)\n\nplt.scatter(x[:,0],x[:,1],c=predict)\nplt.show()\n\n\n可以发现，使用实现的方法进行聚类的结果与真实情况非常吻合。\n"},"Chapter8/":{"url":"Chapter8/","title":"第八章 Apriori","keywords":"","body":"第八章 Apriori\n"},"Chapter8/Apriori算法思想.html":{"url":"Chapter8/Apriori算法思想.html","title":"8.1:Apriori算法思想","keywords":"","body":"8.1:Apriori算法思想\n"},"Chapter8/Apriori算法原理.html":{"url":"Chapter8/Apriori算法原理.html","title":"8.2:Apriori算法原理","keywords":"","body":"8.2:Apriori算法原理\n"},"Chapter8/Apriori算法流程.html":{"url":"Chapter8/Apriori算法流程.html","title":"8.3:Apriori算法流程","keywords":"","body":"8.3:Apriori算法流程\n"},"Chapter8/动手实现Apriori.html":{"url":"Chapter8/动手实现Apriori.html","title":"8.4:动手实现Apriori","keywords":"","body":"8.4:动手实现Apriori\n"},"Chapter8/实战案例.html":{"url":"Chapter8/实战案例.html","title":"8.5:实战案例","keywords":"","body":"8.5:实战案例\n"},"Chapter9/":{"url":"Chapter9/","title":"第九章 PageRank","keywords":"","body":"第九章 PageRank\n"},"Chapter9/PageRank算法思想.html":{"url":"Chapter9/PageRank算法思想.html","title":"9.1:PageRank算法思想","keywords":"","body":"9.1:PageRank算法思想\n"},"Chapter9/PageRank算法原理.html":{"url":"Chapter9/PageRank算法原理.html","title":"9.2:PageRank算法原理","keywords":"","body":"9.2:PageRank算法原理\n"},"Chapter9/PageRank算法流程.html":{"url":"Chapter9/PageRank算法流程.html","title":"9.3:PageRank算法流程","keywords":"","body":"9.3:PageRank算法流程\n"},"Chapter9/动手实现PageRank.html":{"url":"Chapter9/动手实现PageRank.html","title":"9.4:动手实现PageRank","keywords":"","body":"9.4:动手实现PageRank\n"},"Chapter9/实战案例.html":{"url":"Chapter9/实战案例.html","title":"9.5:实战案例","keywords":"","body":"9.5:实战案例\n"},"Chapter10/":{"url":"Chapter10/","title":"第十章 推荐系统","keywords":"","body":"第十章 推荐系统\n"},"Chapter10/推荐系统概述.html":{"url":"Chapter10/推荐系统概述.html","title":"10.1:推荐系统概述","keywords":"","body":"10.1:推荐系统概述\n"},"Chapter10/基于矩阵分解的协同过滤算法思想.html":{"url":"Chapter10/基于矩阵分解的协同过滤算法思想.html","title":"10.2:基于矩阵分解的协同过滤算法思想","keywords":"","body":"10.2:基于矩阵分解的协同过滤算法思想\n在推荐系统中，我们经常看到如下图的表格，表格中的数字代表用户对某个物品的评分，0代表未评分。我们希望能够预测目标用户对物品的评分，进而根据评分高低，将分高的物品推荐给用户。\n\n\n\ny\n物品1\n物品2\n物品3\n物品4\n物品5\n\n\n\n\n用户1\n5\n5\n0\n1\n1\n\n\n用户2\n5\n0\n4\n1\n1\n\n\n用户3\n1\n0\n1\n5\n5\n\n\n用户4\n1\n1\n0\n4\n0\n\n\n\n基于矩阵分解的协同过滤算法正好能解决这个问题。\n基于矩阵分解的协同过滤算法通常都会构造如下图所示评分表y，这里我们以电影为例：\n\n\n\ny\n电影1\n电影2\n电影3\n电影4\n电影5\n\n\n\n\n用户1\n5\n5\n0\n1\n1\n\n\n用户2\n5\n0\n4\n1\n1\n\n\n用户3\n1\n0\n1\n5\n5\n\n\n用户4\n1\n1\n0\n4\n0\n\n\n\n我们认为，有很多因素会影响到用户给电影评分，如电影内容：感情戏，恐怖元素，动作成分，推理悬疑等等。假设我们现在想预测用户2对电影2的评分，用户2他很喜欢看动作片与推理悬疑，不喜欢看感情戏与恐怖的元素，而电影2只有少量的感情戏与恐怖元素，大部分都是动作与推理的剧情，则用户2对电影2评分可能很高，比如5分。\n基于上面的设想，我们只要知道所有用户对电影内容各种元素喜欢程度与所有电影内容的成分，就能预测出所有用户对所有电影的评分了。\n若只考虑两种元素则用户喜好表与电影内容表如下：\n用户喜好表x：\n\n\n\nx\n因素1\n因素2\n\n\n\n\n用户1\n5\n0\n\n\n用户2\n5\n0\n\n\n用户3\n0\n5\n\n\n用户4\n0\n5\n\n\n\n值越大代表用户越喜欢某种元素。\n电影内容表：w:\n\n\n\nw\n电影1\n电影2\n电影3\n电影4\n电影5\n\n\n\n\n因素1\n0.9\n1.0\n0.99\n0.1\n0\n\n\n因素2\n0\n0.01\n0\n1.0\n0.9\n\n\n\n值越大代表电影中某元素内容越多。\n用户2对电影2评分为：5×1.0+0×0.01=5.05\\times 1.0 +0\\times 0.01 = 5.05×1.0+0×0.01=5.0\n对于所有用户，我们可以将矩阵x与矩阵w相乘，得到所有用户对所有电影的预测评分如下表：\n\n\n\nxw\n电影1\n电影2\n电影3\n电影4\n电影5\n\n\n\n\n用户1\n4.5\n5.0\n4.95\n0.5\n0\n\n\n用户2\n4.5\n5.0\n4.95\n0.5\n0\n\n\n用户3\n0\n0.05\n0\n5\n4.5\n\n\n用户4\n0\n0.05\n0\n5\n4.5\n\n\n\n假设电影评分表y（为m行n列的矩阵）,我们考虑d种元素，则电影评分表可以分解为用户喜好表x（为m行d列的矩阵），与电影内容表w（为d行n列的矩阵）。其中d为超参数，大小由我们自己定。\n基于矩阵分解的协同过滤算法思想为：一个用户评分矩阵可以分解为一个用户喜好矩阵与内容矩阵，我们只要能找出正确的用户喜好矩阵参数与内容矩阵参数（即表内的值），就能对用户评分进行预测，再根据预测结果对用户进行推荐。\n"},"Chapter10/基于矩阵分解的协同过滤算法原理.html":{"url":"Chapter10/基于矩阵分解的协同过滤算法原理.html","title":"10.3:基于矩阵分解的协同过滤算法原理","keywords":"","body":"10.3:基于矩阵分解的协同过滤算法原理\n将用户喜好矩阵与内容矩阵进行矩阵乘法就能得到用户对物品的预测结果，而我们的目的是预测结果与真实情况越接近越好。所以，我们将预测值与评分表中已评分部分的值构造平方差损失函数：\nloss=12∑(i,j)∈r(i,j)=1(∑l=1dxilwlj−yij)2\nloss = \\frac{1}{2}\\sum\\limits_{(i,j)\\in r(i,j)=1}(\\sum\\limits_{l=1}^dx_{il}w_{lj}-y_{ij})^2\nloss=21(i,j)∈r(i,j)=1∑(l=1∑dxilwlj−yij)2\n其中：\n\ni:第i个用户\nj:第j个物品\nd:第d种因素\nx:用户喜好矩阵\nw:内容矩阵\ny:评分矩阵\nr:评分记录矩阵，无评分记为0，有评分记为1。r(i,j)=1代表用户i对物品j进行过评分，r(i,j)=0代表用户i对物品j未进行过评分\n\n损失函数python实现代码如下：\nimport numpy as np\nloss = np.mean(np.multiply((y-np.dot(x,w))**2,record))\n\n其中，record为评分记录矩阵。\n我们的目的就是最小化平方差损失函数，通常机器学习都是使用梯度下降的方法来最小化损失函数得到正确的参数。\n对每个参数求得偏导如下：\n∂loss∂xik=∑j∈r(i,j)=1(∑l=1dxilwlj−yij)wkj\n\\frac{\\partial loss}{\\partial x_{ik}} = \\sum\\limits_{j\\in r(i,j)=1}(\\sum\\limits_{l=1}^dx_{il}w_{lj}-y_{ij})w_{kj}\n∂xik∂loss=j∈r(i,j)=1∑(l=1∑dxilwlj−yij)wkj\n∂loss∂wkj=∑i∈r(i,j)=1(∑l=1dxilwlj−yij)xik\n\\frac{\\partial loss}{\\partial w_{kj}} = \\sum\\limits_{i\\in r(i,j)=1}(\\sum\\limits_{l=1}^dx_{il}w_{lj}-y_{ij})x_{ik}\n∂wkj∂loss=i∈r(i,j)=1∑(l=1∑dxilwlj−yij)xik\n则梯度为：\nΔx=r.(xw−y)wT\n\\Delta x = r.(xw-y)w^T\nΔx=r.(xw−y)wT\nΔw=xT[(xw−y).r]\n\\Delta w = x^T[(xw-y).r]\nΔw=xT[(xw−y).r]\n其中：\n.表示点乘法，无则表示矩阵相乘\n上标T表示矩阵转置\n梯度python代码如下：\nx_grads = np.dot(np.multiply(record,np.dot(x,w)-y),w.T)\nw_grads = np.dot(x.T,np.multiply(record,np.dot(x,w)-y))\n\n然后再进行梯度下降:\n#梯度下降，更新参数\nfor i in range(n_iter):\n    x_grads = np.dot(np.multiply(record,np.dot(x,w)-y),w.T)\n    w_grads = np.dot(x.T,np.multiply(record,np.dot(x,w)-y))\n    x = alpha*x - lr*x_grads\n    w = alpha*w - lr*w_grads\n\n其中：\nn_iter:训练轮数\nlr:学习率\nalpha：权重衰减系数，用来防止过拟合\n"},"Chapter10/基于矩阵分解的协同过滤算法流程.html":{"url":"Chapter10/基于矩阵分解的协同过滤算法流程.html","title":"10.4:基于矩阵分解的协同过滤算法流程","keywords":"","body":"10.4:基于矩阵分解的协同过滤算法流程\n\n1.随机初始矩阵值\n2.构造损失函数，求得矩阵参数梯度\n3.进行梯度下降，更新矩阵参数值\n4.喜好矩阵与内容矩阵相乘得到预测评分\n5.根据预测评分进行推荐\n\n"},"Chapter10/动手实现基于矩阵分解的协同过滤.html":{"url":"Chapter10/动手实现基于矩阵分解的协同过滤.html","title":"10.5:动手实现基于矩阵分解的协同过滤","keywords":"","body":"10.5:动手实现基于矩阵分解的协同过滤\n# -*- coding: utf-8 -*-\n\nimport numpy as np\n\ndef recommend(userID,lr,alpha,d,n_iter,data):\n    '''\n    userID(int):推荐用户ID\n    lr(float):学习率\n    alpha(float):权重衰减系数\n    d(int):矩阵分解因子\n    n_iter(int):训练轮数\n    data(ndarray):电影评分表\n    ''' \n    #获取用户数与电影数\n    m,n = data.shape \n    #初始化参数  \n    x = np.random.uniform(0,1,(m,d))\n    w = np.random.uniform(0,1,(d,n))\n    #创建评分记录表，无评分记为0，有评分记为1\n    record = np.array(data>0,dtype=int)\n    #梯度下降，更新参数           \n    for i in range(n_iter):\n        x_grads = np.dot(np.multiply(record,np.dot(x,w)-data),w.T)\n        w_grads = np.dot(x.T,np.multiply(record,np.dot(x,w)-data))\n        x = alpha*x - lr*x_grads\n        w = alpha*w - lr*w_grads\n    #预测\n    predict = np.dot(x,w)\n    #将用户未看过的电影分值从低到高进行排列\n    for i in range(n):\n        if record[userID-1][i] == 1 :\n            predict[userID-1][i] = 0 \n    recommend = np.argsort(predict[userID-1])\n    a = recommend[-1]\n    b = recommend[-2]\n    c = recommend[-3]\n    d = recommend[-4]\n    e = recommend[-5]\n    print('为用户%d推荐的电影为：\\n1:%s\\n2:%s\\n3:%s\\n4:%s\\n5:%s。'\\\n          %(userID,movies_df['title'][a],movies_df['title'][b],movies_df['title'][c],movies_df['title'][d],movies_df['title'][e]))\n\n"},"Chapter10/实战案例.html":{"url":"Chapter10/实战案例.html","title":"10.6:实战案例","keywords":"","body":"10.6:实战案例\n电影评分数据\n本次使用电影评分数据为672个用户对9123部电影的评分记录，部分数据如下：\n\n\n\nuserId\nmovieRow\nrating\n\n\n\n\n1\n30\n2.5\n\n\n7\n30\n3\n\n\n31\n30\n4\n\n\n32\n30\n4\n\n\n\n其中：\nuserId：用户编号\nmovieRow:电影编号\nrating:评分值\n如:\n\n第一行数据表示用户1对电影30评分为2.5分。\n第二行数据表示用户7对电影30评分为3分。\n\n然后，我们还有电影编号与电影名字对应的数据如下：\n\n\n\nmovieRow\ntitle\n\n\n\n\n0\nToy Story (1995)\n\n\n1\nJumanji (1995)\n\n\n2\nGrumpier Old Men (1995)\n\n\n3\nWaiting to Exhale (1995)\n\n\n\n其中：\nmovieRow：电影编号\ntitle：电影名称\n数据下载连接 提取码：ve3v\n构造用户-电影评分矩阵\n大家已经知道，要使用基于矩阵分解的协同过滤算法，首先得有用户与电影评分的矩阵，而我们实际中的数据并不是以这样的形式保存，所以在使用算法前要先构造出用户-电影评分矩阵，python实现代码如下：\nimport numpy as np\n#获取用户数与电影数\nuserNo = max(ratings_df['userId'])+1\nmovieNo = max(ratings_df['movieRow'])+1\n\n#创建电影评分表\nrating = np.zeros((userNo,movieNo))\nfor index,row in ratings_df.iterrows():\n    rating[int(row['userId']),int(row['movieRow'])]=row['rating']\n\n构造出表格后，我们就能使用上一关实现的方法来对用户进行电影推荐了：\nrecommend(1,1e-4,0.999,20,100,rating)\n>>>\n为用户1推荐的电影为：\n1:Rumble Fish (1983)\n2:Aquamarine (2006)\n3:Stay Alive (2006)\n4:Betrayal, The (Nerakhoon) (2008)\n5:Midnight Express (1978)。\n\nrecommend(666,1e-4,0.999,20,100,rating) \n>>>\n为用户666推荐的电影为：\n1:Aquamarine (2006)\n2:It's a Boy Girl Thing (2006)\n3:Kill the Messenger (2014)\n4:Onion Field, The (1979)\n5:Wind Rises, The (Kaze tachinu) (2013)。\n\nrecommend(555,1e-4,0.999,20,100,rating) \n>>>\n为用户555推荐的电影为：\n1:Return from Witch Mountain (1978)\n2:Hitcher, The (2007)\n3:Betrayal, The (Nerakhoon) (2008)\n4:Listen to Me Marlon (2015)\n5:World of Tomorrow (2015)。\n\nrecommend(88,1e-4,0.999,20,100,rating) \n>>>\n为用户88推荐的电影为：\n1:Now, Voyager (1942)\n2:Betrayal, The (Nerakhoon) (2008)\n3:Aquamarine (2006)\n4:Post Grad (2009)\n5:Hitcher, The (2007)\n\n"}}}
						
						
					
				
				
					
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